We define a cutoff value \(c\) and assume that treatment depends on the running variable as follows:

\[ W_i = 1\left(\{Z_i\geq c\}\right), \]

i.e., \(i\) where \(Z_i\) exceeds the cutoff are deemed treated, and all others are deemed not treated.

This treatment assignment is considered as good as random in the close vicinity of the cutoff point.

The following are examples for this kind of treatment assignment:

Why do previously considered approaches not apply in this setting?

Propensity-score methods required two assumptions, unconfoundedness and overlap:

\[ \begin{aligned} &\textcolor{var(--tertiary-color)}{\{Y_i(0),Y_i(1)\}\perp\!\!\!\!\perp W_i\mid Z_i,} \\ &\textcolor{var(--secondary-color)}{0<\mathbb{P}[W_i=1\mid Z_i]<1}. \end{aligned} \]

We thus cannot use methods that rely on division by \(\mathbb{P}[W_i=1\mid Z_i]\). Instead, we need to compare units with \(Z_i\) near the cutoff that are similar to each other, but do not have contiguous distributions.

Let \(\mu_{(w)}(z) = \mathbb{E}[Y_i(w)\mid Z_i=z]\). Then, if both \(\textcolor{var(--primary-color)}{\mu_{(0)}(z)}\) and \(\textcolor{var(--quarternary-color)}{\mu_{(1)}(z)}\) are continuous, we can identify \(\tau_c=\textcolor{var(--quarternary-color)}{\mu_{(1)}(z)}-\textcolor{var(--primary-color)}{\mu_{(0)}(z)}\), via

\[ \tau_c=\textcolor{var(--quarternary-color)}{\underset{z\downarrow c}{\mathrm{lim}}\mathbb{E}[Y_i(w)\mid Z_i=z]}-\textcolor{var(--primary-color)}{\underset{z\uparrow c}{\mathrm{lim}}\mathbb{E}[Y_i(w)\mid Z_i=z]}, \]

We pick a small bandwidth \(h_n \rightarrow 0\) and a symmetric weighting function \(\textcolor{var(--tertiary-color)}{K(\cdot)}\) and fit \(\mu_{(w)}(z)\) via weighted linear regression on each side of the boundary:

where \(a\) and \(\beta_{(w)}\) are nuisance parameters.

Very generally, we can see that under the continuity assumptions mentioned before, the estimator must be consistent for reasonable choices of the bandwidth \(h_n\). To get more specific, we need a more specific smoothness assumption for \(\mu_{(0)}(z)\) and \(\mu_{(1)}(z)\).

We will thus set out to look for different linear estimators. Before, we noted that we could write the local linear estimator as

\[ \hat{\tau}_c = \sum^n_{i=1}\gamma_iY_i, \]

i.e., a linear function of the outcome vector \(Y\).

In a setting with homoskedastic and Gaussian errors, any linear estimator of that form, whose weights \(\gamma_i\) are only functions of the \(Z_i\), satisfies

where \(W_i=1(\{Z_i>c\})\). Thus, any such estimator will be an accurate estimator for \(\tau_c\) if \(\hat{\tau}_c^*(\gamma)\approx\tau_c\) and \(||\gamma||_2^2\) is small.

But, if the curvature of \(\mu_{(w)}(z)\) is still assumed bounded by \(B\), then the bias can be bounded:

\[ \begin{aligned} &\left|\mathrm{Bias}(\hat{\tau}_c(\gamma)\mid\{Z_1,\dots,Z_n\})\right|\leq I_B(\gamma) \\ &I_B(\gamma) = \mathrm{sup}\left\{\sum^n_{i=1}\gamma_i\mu_{W_i}(Z_i)-(\mu_{(1)}(c)-\mu_{(0)}(c)):|\mu''_{(w)}(z)|\leq B\right\}. \end{aligned} \]

Recall that the mean squared error (MSE) of an estimator is the sum of its variance and squared bias. Because the variance does not depend on the conditional response functions, the worst-case MSE of any linear estimator is the sum of its variance and worst-case bias squared:

\[ \mathrm{MSE}(\hat{\tau}_c(\gamma)\mid\{Z_1,\dots,Z_n\})\leq\sigma^2||\gamma||_2^2+I_B^2(\gamma). \]

Thus, assuming that \(|\mu''_{(w)}(z)|\leq B\) and conditionally on \(\{Z_1,\dots,Z_n\}\), the minimax linear estimator is the one that minimizes

\[ \hat{\tau}_c(\gamma^B)=\sum^N_{i=1}\gamma_i^BY_i,\qquad\gamma^B=\mathrm{argmin}\{\sigma^2||\gamma||_2^2+I_B^2(\gamma)\}. \]

We can solve for the weights \(\gamma_i^B\) via quadratic programming.

From before, we know that the errors of our estimator are distributed as

\[ |\mathrm{err} | \{Z_1, ..., Z_n\} \sim \mathcal{N}\bigl(\mathrm{bias}, \sigma^2 \|\gamma^B\|_2^2\bigr). \]

In addition, the optimization procedure from before yields an upper bound for the bias as by-product in terms of the optimization variable \(t\), \(|\mathrm{bias}|\leq Bt\).

We can use the information from the previous slide to build confidence intervals as follows: Because the Gaussian distribution is unimodal and symmetric,

\[ \mathbb{P}[|\mathrm{err}| \ge \zeta] \leq\mathbb{P}[|B\,t + \sigma \|\gamma^B\|_2 \, S| \ge \zeta], \quad S \sim \mathcal{N}(0,1). \]

We can then obtain \(\alpha\)-level confidence intervals like this:

\[ \begin{aligned} &\mathbb{P}[\tau_c\in\mathcal{I}_\alpha\mid\{Z_1,\dots,Z_n\}]\geq 1-\alpha,\\ &\mathcal{I}_\alpha=(\hat{\tau}_c(\gamma^B)-\zeta_\alpha^B,\hat{\tau}_c(\gamma^B)+\zeta_\alpha^B), \\ &\zeta_\alpha^B=\mathrm{inf}\{\zeta:\mathbb{P}[|Bt+\sigma||\gamma^B||_2S|>\zeta]\leq \alpha,\quad S\sim\mathcal{N}(0,1)\}. \end{aligned} \]

These confidence intervals

• account for bias,
• and hold without any distributional assumptions on the running variable because they hold conditionally on \(Z_i\).

If we do not have Gaussian and constant-variance errors, we need to invoke a central limit theorem to argue that

However, if we assume that this approximation is valid, we can still get confidence intervals as before. We can also estimate the conditional variance in the previous equation via

where \(\hat{\mu}_{W_i}(Z_i)\) can, e.g., be derived via local linear regression.

However, the estimator from before is not necessarily minimax under heteroskedasticity. If we use it anyways, we can build confidence intervals using the procedure on this slide, but should be aware that the estimator is motivated by a more simplified model.